Signals and Systems - Electrical Engineering

116 C H A P T E R 1: Continuous-Time Signals

sum of these sinusoids. Assume theNPplayers while trying to play the pure tone end up playing tones

separated by0.02Hz, so that the recorded signal is

y(t)=

∑NP

i= 1

10 cos( 2 πfit)

where thefiare frequencies from 159 to 161 separated by 1 Hz.

(a) Generate the signaly(t) 0 ≤t≤ 200 sec in MATLAB. Let each musician play a unique frequency.

Consider an increasing number of players, lettingNPbe first 51 players with 1 =0.04 Hz, and then

101 players with 1 =0.02 Hz. Ploty(t)for each of the different number of players.

(b)Explain how this is related with multipath and Doppler effects discussed in the previous problems.

1.23. Chirps—MATLAB

Pure tones or sinusoids are not very interesting to listen to. Modulation and other techniques are used to

generate more interesting sounds. Chirps, which are sinusoids with time-varying frequency, are some of

those more interesting sounds. For instance, the following is a chirp signal:

y(t)=Acos(ct+s(t))

(a) LetA= 1 ,c= 2 , ands(t)=t^2 / 4. Use MATLAB to plot this signal for 0 ≤t≤40 secin steps of

0.05 sec. Use thesoundfunction to listen to the signal.

(b)LetA= 1 ,c= 2 , ands(t)=−2 sin(t). Use MATLAB to plot this signal for 0 ≤t≤40 secin steps of

0.05 sec. Use thesoundfunction to listen to the signal.

(c)The frequency of these chirps is not clear. The instantaneous frequencyIF(t)is the derivative with

respect totof the argument of the cosine. For instance, for a cosinecos( 0 t), theIF(t)=d 0 t/dt=

 0 , so that the instantaneous frequency coincides with the conventional frequency. Determine the

instantaneous frequencies of the two chirps and plot them. Do they make sense as frequencies?

Explain.